On uniqueness sets of additive eigenvalue problems and applications
aa r X i v : . [ m a t h . A P ] J a n ON UNIQUENESS SETS OF ADDITIVE EIGENVALUEPROBLEMS AND APPLICATIONS
HIROYOSHI MITAKE AND HUNG V. TRAN
Abstract.
In this paper, we provide a simple way to find uniqueness sets foradditive eigenvalue problems of first and second order Hamilton–Jacobi equationsby using a PDE approach. An application in finding the limiting profiles for largetime behaviors of first order Hamilton–Jacobi equations is also obtained. Introduction
Let T n be the usual n -dimensional torus. Let the Hamiltonian H = H ( x, p ) ∈ C ( T n × R n ) be such that(H1) for every x ∈ T n , p H ( x, p ) is convex,(H2) uniformly for x ∈ T n ,lim | p |→∞ H ( x, p ) | p | = + ∞ and lim | p |→∞ (cid:18) H ( x, p ) + D x H ( x, p ) · p (cid:19) = + ∞ . The first order additive eigenvalue (ergodic) problem corresponding to H is(E) H ( x, Dw ) = c in T n . Here, ( w, c ) ∈ C ( T n ) × R is a pair of unknowns. It was shown in [11] that thereexists a unique constant c ∈ R such that (E) has a viscosity solution w ∈ C ( T n ).We denote by c the ergodic constant of (E). Without loss of generality, we normalizethe ergodic constant c to be zero henceforth.One of the most interesting points to study (E) is that (E) is not monotone in w , and in general, (E) has many viscosity solutions of different types (see examplesin [10, Chapter 6] for instance). It is therefore fundamental to understand why this nonuniqueness phenomenon appears, and in particular, to find a uniqueness set for(E). It turns out that this has deep relations to Hamiltonian dynamics and weakKAM theory. In fact, a uniqueness set for (E) has already been studied in [7, 8] inthe context of weak KAM theory.In this short paper, we provide a new and simple way to look at this phenomenonby using PDE techniques. Some applications and generalizations are also provided. Mathematics Subject Classification.
Key words and phrases.
Uniqueness set; Hamilton–Jacobi equations; Mather measures; Non-linear adjoint methods.The work of HM was partially supported by the JSPS grants: KAKENHI
Settings and main results.
We first recall the definition of Mather measures.Consider the following minimization problemmin µ ∈F Z Z T n × R n L ( x, v ) dµ ( x, v ) , (1.1)where L is the Legendre transform of H , that is, L ( x, v ) = sup p ∈ R n ( p · v − H ( x, p )) for ( x, v ) ∈ T n × R n , and F = (cid:26) µ ∈ P ( T n × R n ) : Z Z T n × R n v · Dφ ( x ) dµ ( x, v ) = 0 for all φ ∈ C ( T n ) (cid:27) . Here, P ( T n × R n ) is the set of all Radon probability measures on T n × R n . Measuresbelong to F are called holonomic measures associated with (E). Definition 1 (Mather measures) . Let f M ⊂ F be the set of all minimizers of (1.1) .Each measure in f M is called a Mather measure. As we normalize c = 0, we actually have that (see [13, 12, 7, 8] for instance)min µ ∈F Z Z T n × R n L ( x, v ) dµ ( x, v ) = − c = 0 . (1.2)See [14], [10, Lemma 6.12] for a proof of a more general version this fact. Here isour first main result. Theorem 1.1.
Assume (H1)–(H2) . Let w , w be two viscosity solutions of ergodicproblem (E) . Assume further that Z Z T n × R n w ( x ) dµ ( x, v ) ≤ Z Z T n × R n w ( x ) dµ ( x, v ) for all µ ∈ f M . Then w ≤ w in T n . Let M be the projected Mather set on T n , that is, M = [ µ ∈ f M supp (proj T n µ ) . Theorem 1.1 gives the following straightforward result.
Corollary 1.2.
Assume (H1)–(H2) . Let w , w be two viscosity solutions of ergodicproblem (E) . Assume further that w = w on M . Then w = w in T n . Corollary 1.2 was derived in [7, 8] much earlier. We provide a simple prooffor Theorem 1.1 in Section 2, which is a new application of the nonlinear adjointmethod introduced in [5] (see also [15]). A generalization of Theorem 1.1 to thesecond order (degenerate viscous) setting, Theorem 4.1, is given in Section 4. It isworth mentioning that the result of Theorem 4.1 is new in the literature.
NIQUENESS SET 3
Application.
We provide here an application in large time behavior. In thiscontext, we need to strengthen the convexity of H in (H1).(H1’) There exists γ > D pp H ( x, p ) ≥ γI n for all ( x, p ) ∈ T n × R n . Here, I n is the identity matrix of size n .Under assumptions (H1’), (H2) and that the ergodic constant c = 0, for given u ∈ Lip ( T n ), the viscosity solution u ∈ C ( T n × [0 , ∞ )) of the Cauchy problem(C) ( u t + H ( x, Du ) = 0 in T n × (0 , ∞ ) ,u ( x,
0) = u ( x ) on T n . has the following large time behaviorlim t →∞ k u ( · , t ) − v k L ∞ ( T n ) = 0 , (1.3)where v ∈ Lip ( T n ) is a viscosity solution of (E). This result was first proved in[6]. Notice that there are various different ways to prove it (see [2, 3, 10] and thereferences therein). We say that v is the asymptotic profile of u , and denote it by u ∞ , or u ∞ [ u ] to display the clear dependence on the initial data u .We now give a representation formula for u ∞ [ u ]. Theorem 1.3 (Asymptotic profiles) . Assume that (H1’) and (H2) hold, and theergodic constant c = 0 . For given u ∈ Lip ( T n ) , let u ∞ [ u ] be the correspondingasymptotic profile. Then, we have (i) u ∞ [ u ]( y ) = u − ( y ) for all y ∈ M , (ii) u ∞ [ u ]( x ) = min (cid:8) d ( x, y ) + u − ( y ) : y ∈ M (cid:9) for all x ∈ T n .Here, u − ( x ) = sup { v ( x ) : v ≤ u on T n , and v is a subsolution to (E) } ,d ( x, y ) = sup { v ( x ) − v ( y ) : v is a subsolution to (E) } . Theorem 1.3 was first proved in [4, Theorem 3.1], and our purpose is to give adifferent proof in Section 3, which seems to be simpler.2.
Uniqueness set of the ergodic problem
We present in this section the proof of Theorem 1.1.
Proof of Theorem . We use ideas introduced in [3].For each i = 1 , ε >
0, let u εi be the viscosity solution to the Cauchyproblem ( ε ( u εi ) t + H ( x, Du εi ) = ε ∆ u εi in T n × (0 , ,u εi ( x,
0) = w i ( x ) on T n . (2.1)Without the viscosity term, (2.1) becomes ( ε ( u i ) t + H ( x, Du i ) = 0 in T n × (0 , ,u i ( x,
0) = w i ( x ) on T n . (2.2) H. MITAKE, H. V. TRAN
It is clear that the unique viscosity solution to (2.2) is u i ( x, t ) = w i ( x ) for all( x, t ) ∈ T n × [0 ,
1) because of the fact that w i is a viscosity solution to (E). Thanksto (H2), by a standard argument, there exists C > ε such that k Du εi k L ∞ ( T n × (0 , ≤ C (2.3)and k u εi − w i k L ∞ ( T n × (0 , ≤ Cε. (2.4)See [10, Propositions 4.15 and 5.5] for the proofs of similar versions of (2.3) and(2.4) for instance. Our plan is to use u ε , u ε to deduce the conclusion as ε → x ∈ T n , let σ ε be the solution to ( − εσ εt − div( D p H ( x, Du ε ) σ ε ) = ε ∆ σ ε in T n × (0 , ,σ ε ( x,
1) = δ x on T n . Here δ x is the Dirac delta mass at x .By convexity of H in (H1), we have ε ( u ε − u ε ) t + D p H ( x, Du ε ) · D ( u ε − u ε ) ≤ ε ∆( u ε − u ε ) . Multiply this by σ ε , integrate on T n , and note that Z T n (cid:0) − D p H ( x, Du ε ) · D ( u ε − u ε ) + ε ∆( u ε − u ε ) (cid:1) σ ε dx = Z T n (cid:0) div( D p H ( x, Du ε ) σ ε ) + ε ∆ σ ε (cid:1) ( u ε − u ε ) dx = − Z T n εσ εt ( u ε − u ε ) dx. Thus, ddt Z T n ( u ε − u ε ) σ ε dx ≤ , which yields ( u ε − u ε )( x , ≤ Z Z T n ( u ε − u ε ) σ ε dxdt. (2.5)In light of the Riesz theorem, there exists ν ε ∈ P ( T n × R n ) such that Z Z T n × R n ϕ ( x, p ) dν ε ( x, p ) = Z Z T n ϕ ( x, Du ε ) σ ε dxdt for all ϕ ∈ C c ( T n × R n ) . (2.6)Then, (2.5) becomes( u ε − u ε )( x , ≤ Z Z T n × R n ( u ε − u ε ) dν ε ( x, p ) . (2.7)Thanks to (2.3), we have that supp( ν ε ) ⊂ T n × B (0 , C ). There exists { ε j } → ν ε j ⇀ ν ∈ P ( T n × R n ) as j → ∞ weakly in the sense of measures. Weset µ ∈ P ( T n × R n ) be such that Z Z T n × R n ϕ ( x, p ) dν ( x, p ) = Z Z T n × R n ϕ ( x, D v L ( x, v )) dµ ( x, v ) . (2.8)We provide a proof that µ is a Mather measure in Lemma 2.1 below for completeness(see also [14, Proposition 2.3], [10, Proposition 6.11]). NIQUENESS SET 5
Sending j → ∞ in (2.7) and using (2.4) to yield w ( x ) − w ( x ) ≤ Z Z T n × R n ( w − w ) dµ ( x, v ) ≤ . (cid:3) Lemma 2.1.
For each ε > , let ν ε be the measure defined in (2.6) . Assume thatthere exists a sequence { ε j } → such that ν ε j ⇀ ν ∈ P ( T n × R n ) as j → ∞ weaklyin the sense of measures. Let µ be a measure defined through ν by (2.8) . Then µ isa Mather measure.Proof. Fix any φ ∈ C ( T n ), and consider a family { φ m } ⊂ C ∞ ( T n ) such that φ m → φ in C ( T n ) as m → ∞ .Multiply the adjoint equation with φ m and integrate on T n × [0 ,
1] to imply ε Z T n φ m ( x ) σ ε ( x, dx − εφ m ( x ) + Z Z T n D p H ( x, Du ε ) · Dφ m ( x ) σ ε ( x, t ) dxdt = ε Z Z T n ∆ φ m ( x ) σ ε ( x, t ) dxdt. Let ε = ε j → m → ∞ in this order to get Z Z T n × R n D p H ( x, p ) · Dφ ( x ) dν ( x, p ) = Z Z T n × R n v · Dφ ( x ) dµ ( x, v ) = 0 . Thus, µ ∈ F .We rewrite (2.1) as ε ( u ε ) t + D p H ( x, Du ε ) · Du ε − ε ∆ u ε = D p H ( x, Du ε ) · Du ε − H ( x, Du ε ) . Multiply this by σ ε and integrate on T n × [0 ,
1] to yield εu ε ( x , − ε Z T n u ε ( x, σ ε ( x, dx = Z Z T n ( D p H ( x, Du ε ) · Du ε − H ( x, Du ε )) σ ε dxdt. We again let ε = ε j → Z Z T n × R n ( D p H ( x, p ) · p − H ( x, p )) dν ( x, p ) = Z Z T n × R n L ( x, v ) dµ ( x, v ) . Also, note that we have
Z Z T n × R n L ( x, v ) dµ ≥ µ ∈ F , (2.9)which, together with (1.2), completes the proof. See [10, Lemma 6.12] for a proofof (2.9). (cid:3) Application
In this section, we always assume that (H1’)–(H2) hold and that the ergodicconstant c = 0. Lemma 3.1.
Assume that u is a viscosity subsolution of (E) . Then, u ∞ [ u ] = u on M . H. MITAKE, H. V. TRAN
Proof.
We write u ∞ for u ∞ [ u ] in the proof for simplicity.By the usual comparison principle, we have u ( x, t ) ≥ u ( x ) for all ( x, t ) ∈ T n × [0 , ∞ ). Hence, u ∞ ≥ u on T n .Next, let ρ be a standard mollifier in R n . For each δ >
0, let ρ δ ( x ) = δ − n ρ ( δ − x )for all x ∈ R n . Let u δ = ρ δ ∗ u . Then due to the convexity of H in p , u δ is asubsolution to u δt + H ( x, Du δ ) ≤ Cδ in T n × (0 , ∞ ) . For any Mather measure µ ∈ f M , by the holonomic and minimizing properties, wehave ddt Z Z T n × R n u δ ( x, t ) dµ = Z Z T n × R n ( u δt + v · Du δ − L ( x, v )) dµ ≤ Z Z T n × R n u δt + H ( x, Du δ ) dµ ≤ Cδ.
Therefore, for any
T > Z Z T n × R n u δ ( x, T ) dµ ≤ Z Z T n × R n ( u ) δ ( x ) dµ + CδT.
Let δ → T → ∞ in this order to yield Z Z T n × R n u ∞ dµ ≤ Z Z T n × R n u dµ. Combined with u ∞ ≥ u on T n , we obtain u ∞ = u on M , which completes theproof. (cid:3) Remark 1.
Notice that we get u ( x, t ) = u ( x ) for all x ∈ M , t ∈ [0 , ∞ ) , in the above proof.We present next the proof of Theorem 1.3. Before proceeding to the proof, it isimportant noticing that d has the following representation formula d ( x, y ) = inf (cid:26)Z t L ( γ ( s ) , − ˙ γ ( s )) ds : t > , γ ∈ AC ([0 , t ] , T n ) , γ (0) = x, γ ( t ) = y (cid:27) . Proof of Theorem . It is enough to give only the proof of (i). The second claim(ii) follows immediately from Corollary 1.2, claim (i) and the representation formulasof d as well as of solutions to (E).By the definition of u − , we have u − ≤ u on T n . In light of the comparisonprinciple, u − ≤ u on T n × [0 , ∞ ), which implies u − ≤ u ∞ on T n .We prove the reverse inequality holds on M . Fix y ∈ M and z ∈ T n . Set w z ( x ) = u ( z ) + d ( x, z ) for x ∈ T n . Then, note that w z is a viscosity subsolution to(E). Let w be the solution to (C) with initial data w z . Thanks to Lemma 3.1, weget w ( y, t ) = w z ( y ) = u ( z ) + d ( y, z ) for all t ∈ [0 , ∞ ) . (3.1) NIQUENESS SET 7
For a large t >
1, pick γ : [0 , t ] → T n to be an optimal path such that γ (0) = y and w ( y, t ) = w z ( γ ( t ))+ Z t L ( γ ( s ) , − ˙ γ ( s )) ds = u ( z )+ d ( γ ( t ) , z )+ Z t L ( γ ( s ) , − ˙ γ ( s )) ds. On the other hand, for any ε >
0, there exists t ε > γ : [ t, t + t ε ] → T n with γ ( t + t ε ) = z satisfying d ( γ ( t ) , z ) ≥ Z t + t ε t L ( γ ( s ) , − ˙ γ ( s )) ds − ε. Combine the two relations above to imply w z ( y ) + ε ≥ u ( z ) + Z t + t ε L ( γ ( s ) , − ˙ γ ( s )) ds ≥ u ( y, t + t ε ) . (3.2)By letting t → ∞ in (3.2), one gets w z ( y ) + ε ≥ u ∞ ( y ) . Next, let ε → u ( z ) + d ( y, z ) ≥ u ∞ ( y ). Vary z to yield u ∞ ( y ) ≤ min z ∈ T n ( u ( z ) + d ( y, z )) . Notice here that in view of the inf-stability of viscosity subsolutions to convex firstorder Hamilton–Jacobi equations, we have min z ∈ T n ( u ( z ) + d ( y, z )) = u − ( y ), whichfinishes the proof. (cid:3) Generalization: degenerate viscous cases
In this section, we present a generalization of Theorem 1.1 to the second order(degenerate viscous) setting. In this setting, the ergodic problem is(VE) H ( x, Dw ) = tr (cid:0) A ( x ) D w (cid:1) + c in T n . As above, ( w, c ) ∈ C ( T n ) × R is a pair of unknowns. Here A : T n → M n × n sym is thediffusion matrix, where M n × n sym is the set of all n × n real symmetric matrices. Weneed the following assumptions.(H2’) There exist γ > C > x, p ) ∈ T n × R n , C | p | γ − C ≤ H ( x, p ) ≤ C ( | p | γ + 1) , | D x H ( x, p ) | ≤ C (1 + | p | γ ) , | D p H ( x, p ) | ≤ C (1 + | p | γ − ) . (H3) A ( x ) = ( a ij ( x )) ≤ i,j ≤ n ∈ M n × n sym with A ≥
0, and a ij ∈ C ( T n ) for all 1 ≤ i, j ≤ n .By normalization, we always assume that c = 0 in this section. In fact, underassumptions (H1), (H2’) and (H3), for any w ∈ C ( T n ) solving (VE), w ∈ Lip ( T n )(see [1, Theorem 3.1]). H. MITAKE, H. V. TRAN
Definition 2.
Let f M V be the set of all minimizers of the minimizing problem min µ ∈F Z Z T n × R n L ( x, v ) dµ ( x, v ) , (4.1) where F V = (cid:26) µ ∈ P ( T n × R n ) : Z Z T n × R n v · Dφ − a ij φ x i x j dµ ( x, v ) = 0 for all φ ∈ C ( T n ) (cid:27) . Each measure in f M V is called a generalized Mather measure. Because of normalization that c = 0, as in the first order case, one has thatmin µ ∈F V Z Z T n × R n L ( x, v ) dµ ( x, v ) = 0 . (4.2)The proof of this claim follows [10, Lemma 6.12]. To be more precise, [10, Lemma6.12] deals with the special case A ( x ) = a ( x ) I n where a ∈ C ( T n , [0 , ∞ )) and I n is the identity matrix of size n . For general diffusion matrix A satisfying (H3), weperform first inf-sup convolutions, and then normal convolution of a solution w of(VE). See also [9] for a form of (4.2) in fully nonlinear, degenerate elliptic PDEsettings. Theorem 4.1.
Assume (H1), (H2’), (H3) . Let w , w be two continuous viscositysolutions of ergodic problem (E) . Assume further that Z Z T n × R n w ( x ) dµ ( x, v ) ≤ Z Z T n × R n w ( x ) dµ ( x, v ) for all µ ∈ f M V . Then w ≤ w in T n .Proof. We basically repeat the proof of Theorem 1.1.For each k = 1 , ε >
0, let u εk be the solution to the Cauchy problem ( ε ( u εk ) t + H ( x, Du εk ) = a ij ( u εk ) x i x j + ε ∆ u εk in T n × (0 , ,u εk ( x,
0) = w k ( x ) on T n . Without the viscosity ε ∆ u εk , (2.1) becomes ( ε ( u k ) t + H ( x, Du k ) = a ij ( u k ) x i x j in T n × (0 , ,u k ( x,
0) = w k ( x ) on T n , (4.3)It is clear that the unique viscosity solution to (4.3) is u k ( x, t ) = w k ( x ) for all( x, t ) ∈ T n × [0 ,
1) because of the fact that w k is a solution to (VE). Thanks to(H2’) (see [10, Theorem 4.5] for instance), there exists C > ε suchthat k Du εi k L ∞ ( T n × (0 , ≤ C and k u εi − w i k L ∞ ( T n × (0 , ≤ Cε. (4.4)As above, we use u ε , u ε to deduce the conclusion as ε → x ∈ T n , let σ ε be the solution to ( − εσ εt − div( D p H ( x, Du ε ) σ ε ) = ( a ij σ ε ) x i x j + ε ∆ σ ε in T n × (0 , ,σ ε ( x,
1) = δ x on T n . NIQUENESS SET 9
Here δ x is the Dirac delta mass at x .By convexity of H , we have ε ( u ε − u ε ) t + D p H ( x, Du ε ) · D ( u ε − u ε ) ≤ a ij ( u ε − u ε ) x i x j + ε ∆( u ε − u ε ) . Multiply this by σ ε and integrate on T n to yield ddt Z T n ( u ε − u ε ) σ ε dx ≤ . Hence, ( u ε − u ε )( x , ≤ Z Z T n ( u ε − u ε ) σ ε dxdt. (4.5)Let ν ε ∈ P ( T n × R n ) be the measure satisfying Z Z T n × R n ϕ ( x, p ) dν ε ( x, p ) = Z Z T n ϕ ( x, Du ε ) σ ε dxdt for all ϕ ∈ C c ( T n × R n ) . Then, (4.5) becomes( u ε − u ε )( x , ≤ Z Z T n × R n ( u ε − u ε ) dν ε ( x, p ) . (4.6)Thanks to (4.4), we have that supp( ν ε ) ⊂ T n × B (0 , C ). There exists { ε j } → ν ε j ⇀ ν ∈ P ( T n × R n ) as j → ∞ weakly in the sense of measures. Weset µ ∈ P ( T n × R n ) be such that Z Z T n × R n ϕ ( x, p ) dν ( x, p ) = Z Z T n × R n ϕ ( x, D v L ( x, v )) dµ ( x, v ) . Note that µ is a generalized Mather measure defined in Definition 2. We refer to[14, Proposition 2.3] or [10, Proposition 6.11] for the details.Sending j → ∞ in (4.6) and using (4.4) to yield w ( x ) − w ( x ) ≤ Z Z T n × R n ( w − w ) dµ ( x, v ) ≤ . (cid:3) Let M V be the generalized projected Mather set on T n , that is, M V = [ µ ∈ f M V supp (proj T n µ ) . Theorem 4.1 gives the following straightforward result.
Corollary 4.2.
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E-mail address : [email protected] (H. V. Tran) Department of Mathematics, University of Wisconsin Madison, 480Lincoln Drive, Madison, WI 53706, USA
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